Partitioning Orthogonal Histograms into Rectangular Boxes
The problem of partitioning an orthogonal polyhedron into a minimum number of boxes was shown to be NP-hard in 1991, but no approximability result is known except for a 4-approximation algorithm for 3D-histograms. In this paper we broaden the understanding of the 3D-histogram partitioning problem. We prove that partitioning a 3D-histogram into a minimum number of boxes is NP-hard, even for histograms of height two. This settles an open question posed by Floderus et al. We then show the problem to be APX-hard for histograms of height four. On the positive side, we give polynomial-time algorithms to compute optimal or approximate box partitions for some restricted but interesting classes of polyhedra and 3D-histograms.
This work was done as part of the Algorithms Problem Session at the University of Waterloo. We thank the other participants for valuable discussions. Research of T.B. and A.L. supported by NSERC, M.D. supported by Vanier CGS, M.D. and D.M. supported by an NSERC PDF.
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